Test 1 Definitions

Question 1

Define a set that is bounded above, below, and what it means for a set S to be bounded:

Answer 1

Definition 1.1 Let S be a nonempty subset of R. Then:
1. If there is a number A ∈ R such that x ≤ A for all x ∈ S, then A is said to be an upper bound of S, and S is said to be bounded above.

2. If there is a number a ∈ R such that x ≥ a for all x ∈ S, then a is said to be a lower bound of S, and S is said to be bounded below.
3. If S is bounded above and bounded below, then S is called bounded.

Question 2

Define the maximum and minimum element of a nonempty subset of R

Answer 2

Definition 1.2 Let S be a nonempty subset of R.
1. If S has an upper bound M which is an element of S, then M is called the greatest element or maximum of S, and we write M = max S.
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2. If S has a lower bound m which is an element of S, then m is called the least element or minimum of S, and we write m = min S.

Question 3

Define the supremum of a nonempty subset of R:

Answer 3

Definition 1.3 Let S be a nonempty subset of R.
A real number M is said to be the supremum or least upper bound of S, if
(a) M is an upper bound of S, and
(b) if L is any upper bound of S, then M ≤ L.
The supremum of S is denoted by sup S.

Question 4

Define the infimum of a nonempty subset of R:

Answer 4

Definition 1.4 Let S be a nonempty subset of R.
A real number m is said to be the inmum or greatest lower bound of S, if
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(a) m is a lower bound of S, and
(b) if l is any lower bound of S, then m ≥ l.
The infimum of S is denoted by inf S.

Question 5

What is the positive square root theorem?

Answer 5

Theorem 1.5 (Positive square root) Let a ≥ 0. Then there is a unique x ≥ 0 such that x^2 = a. We write x =√a = a^(1/2)

Question 6

Define the Dedekind cut:

Answer 6

Theorem 1.9 (Dedekind cut) Let A and B be nonempty subsets of R such that
(i) A ∩ B = ∅,
(ii) A ∪ B = R,
(iii) ∀ a ∈ A∀ b ∈ B; a ≤ b.
Then there is c ∈ R such that a ≤ c ≤ b for all a ∈ A and b ∈ B.

Question 7

Define the archimedan principle:

Answer 7

Theorem 1.10 (The Archimedean principle) For each x ∈ R there is n ∈ N such that n > x.

Question 8

Define what it means for a subset of R to be dense:

Answer 8

Definition 1.5 A subset S of R is said to be dense in R if for all x; y ∈ R with x < y
there is s ∈ S such that x < s < y.